Theorem nnssn0s 28594

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28588	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4086	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3980	1 ⊢ ℕs ⊆ ℕ0s

This proof depends on syntax axioms:   ∖ cdif 3899   ⊆ wss 3902  {csn 4587   0_s c0s 28078  ℕ_0scn0s 28585  ℕ_scnns 28586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-ss 3919  df-nns 28588

This theorem is used by:  nnssno  28595  nnn0s  28600  nnn0sd  28601  nnsgt0  28612